Rectangular band
= Rectangular band
{title2=$x*x=x,\quad(x*y)*(z*w)=x*w$}
A rectangular band is a <semigroup> satisfying the displayed identities; these also imply associativity. Every nonempty such algebra is a product $B\times C$ with $(b,c)*(b',c')=(b,c')$. Fixing $e$, the factor sets are $B=\{x*e\}$ and $C=\{e*x\}$, and $x\mapsto(x*e,e*x)$ has inverse $(b,c)\mapsto b*c$. Homomorphisms between nonempty products are exactly pairs of maps on the two factors.